2Loc QuillenEquivs(CombinatorialModelCategories) ?2Ho(PresentableCategories) \begin{aligned} & 2Loc_{QuillenEquivs} \big( CombinatorialModelCategories ) \\ & \; \overset{\color{purple}?}{\simeq} \; 2Ho \big( Presentable \infty Categories \big) \end{aligned}

For H\mathbf{H} an \infty-topos and GGroups(H)G \in Groups(\mathbf{H}), one would expect that

(1) group objects Γ//GGroups(H /BG)\Gamma /\!\!/ G \,\in\, \mathrm{Groups}\big( \mathbf{H}_{/\mathbf{B}G} \big) in the slice over the delooping of GG

are equivalent to

(2) group objects ΓGroups(H)\Gamma \,\in\, \mathrm{Groups}(\mathbf{H}) equipped with an action of GG by group automorphisms.

A construction (2)(1)(2)\Rightarrow (1) is in Prop. 2.102 on p. 35 of Proper Orbifold Cohomology.

The following are some thoughts on how to go about (1)(2)(1) \Rightarrow (2), but not conclusive yet.

The following pasting diagram shows something slightly weaker:


(a) is the homotopy pullback that exhibits Γ//G\Gamma /\!\!/ G as a group object in the slice over BG\mathbf{B}G;

(b) is the homotopy pullback that exhibits Γ//G\Gamma /\!\!/ G as the homotopy quotient of a GG-action on Γ\Gamma;

(c) is the homotopy fiber product which exhibits the shear map equivalence of Γ\Gamma as a principal GG-bundle over Γ//G\Gamma /\!\!/ G;

(d) is a homotopy pullback implied from this by the pasting law.

While we can’t seem to conclude group structure on Γ\Gamma directly, we see that Γ×G\Gamma \times G carries a group structure, to be denoted ΓG\Gamma \rtimes G, whose delooping is the bottom right object.

Moreover, G(e,id)Γ×GG \xrightarrow{(e,\mathrm{id})} \Gamma \times G is exhibited as a homomorphism of group objects.

Also, we see that G(e,id)Γ×GρΓG \xrightarrow{(e,\mathrm{id})} \Gamma \times G \xrightarrow{\rho} \Gamma are GG-equivariant maps for GG acting by right multiplication on itself. This implies that ρ\rho is the given GG-action,

So we are beginning to see that ΓGGroups(ΓG)\Gamma \!\sslash\! G \in Groups\big( \Gamma \rtimes G\big) is equivalent to a “semidirect product \infty-group” ΓG\Gamma \rtimes G.

But can we make explicit that the action of GG on Γ\Gamma is by group automorphisms? This would require constructing a homotopy fiber sequence of the form

BΓ B(ΓG) BG \array{ \mathbf{B}\Gamma &\longrightarrow& \mathbf{B}(\Gamma \rtimes G) \\ && \big\downarrow \\ && \mathbf{B}G }

How to see this homotopy fiber from the above diagram (or from some other consideration)?

Last revised on June 12, 2021 at 07:19:18. See the history of this page for a list of all contributions to it.